We investigate the Hamilton-Jacobi equation of a probe particle moving on
    d-dimensional generalized Lense-Thirring metric. This space-time is different
    from the slowly rotating Myers-Perry black hole at second order in rotation
    parameters. We show that the dynamics of the probe particle along the time-like
    geodesic of the generalized Lense-Thirring space-time is super-integrable and
    has more constants of motion with respect to the same dynamics on Myers-Perry
    black hole. We also discuss the second rank Killing tensors associated with the
    generalized Lense-Thirring metric.

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